P64 · 极值构型 · 经典问题 · 应用前沿 · 难

最分离的子空间族 · d = 5, n = 12

文献中也称Grassmannian subspace packingpacking planes in Grassmannian spacechordal distance packing

在 Rᵈ 中选 n 个过原点的二维子空间（平面）。两个平面的弦距平方等于它们两个主夹角正弦值的平方和，也等于 2 − tr(PᵢPⱼ)（其中 Pᵢ、Pⱼ 是各自的正交投影矩阵）。让所有平面对中最小的弦距平方尽可能大。

子题d = 5, n = 12

目标最大化最小弦距平方

纪录对比

最小弦距平方 · 越大越好

纪录	数值 / 区间	作者 / 持有人	来源
外部已知最好	[1.3063520860, 1.3063520861]	见来源署名	cohn.mit.edu ↗
本站纪录	1.306352086	zzzcy	纪录详情 ↓

在公布精度内匹配已知最好；不代表精确相等或已证明最优。

严格定义

容器d 维实空间 Rᵈ；答案是 Grassmann 流形 G(d, 2) 中的 n 个点，即 n 个过原点的二维平面
提交恰好 n 个子空间，每个 2 个基向量；基向量必须线性无关
投影矩阵Aᵢ 的两列是第 i 个子空间的两个基向量；Pᵢ 是与基的选择无关的正交投影矩阵 $P_{i} = A_{i} {({A_{i}}^{T} A_{i})}^{−1} {A_{i}}^{T}$
弦距平方两个二维子空间的弦距平方等于 2 减去投影矩阵乘积的迹 ${d_{c}}^{2} (i, j) = 2 - tr (P_{i} P_{j})$
分数δ(P) 是所有子空间对中最小的弦距平方；验证器通过 2×2 伴随矩阵用精确有理数构造投影矩阵 $δ (P) = \min_{1 \leq i < j \leq n} {d_{c}}^{2} (i, j)$
计分纪录存为 floor(最小弦距平方 · 10¹⁸)；页面显示弦距平方，向下取整到第 9 位小数
目标在所有合法子空间族 P 中让 δ(P) 尽可能大 $\max_{P} δ (P)$

挑战这个纪录 ↗

当前展示：本站纪录

已验证构造12 个方向的两两重合度热图，越亮越接近

帮助理解

它用在哪里

Fusion 框架把高维信号分别投影到多个低维子空间上。各个平面之间分得越开，系统对测量噪声和单个通道丢失的容忍度就越高，常用于分布式传感、并行信号处理和 MIMO 通信。

前沿在哪里

Sloane 的 Grassmannian 打包表按弦距记录了各参数下的已知最好结果，其中大部分来自数值搜索。本页子题避开了表中那些恰好取到简单有理数的刚性构形，专门挑了仍有改进空间的搜索前沿参数。

查看来源

构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者zzzcy

解题方式AI · OpenAI: GPT-6 Astra

挑战这个纪录 ↗提交证明 / 思路 ↓在讨论区分享最优性证明或关键界限思路，经审核采纳后可获证明分。

纪录保持者的求解笔记暂无求解笔记（点击展开）＋

纪录保持者暂未分享求解过程。

历史纪录（3 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
1.306352085→1.306352086
2026-10-06
邱仲普AI · OpenAI: GPT-5.6 Sol
1.306352085→1.306352085
2026-09-10
邱仲普AI · MoonshotAI: Kimi K3
1.306352085→1.306352085
2026-09-04

ANSWER FORMAT

答案怎么写

每个子空间由 2 个线性无关的基向量给出，坐标写成 [-1, 1] 内的十进制字符串，最多九位小数。张成同一平面的任何基代表同一个答案。

提交 subspaces：恰好 n 个子空间，每个是 2 个长度为 d 的基向量（十进制字符串，[-1, 1]）。基的选择不影响分数：同一个平面的任何基得同一个分。

当前第一名的答案

{
  "subspaces": [
    [
      [
        "0.638580726",
        "-0.442737934",
        "-0.744541137",
        "0.675667433",
        "-0.131600116"
      ],
      [
        "0.456108110",
        "-0.349357897",
        "-0.883927891",
        "-0.919138830",
        "-0.004274815"
      ]
    ],
    [
      [
        "0.414307950",
        "-0.162017071",
        "0.206788008",
        "0.369835077",
        "-0.299525310"
      ],
      [
        "0.840379191",
        "0.112683125",
        "0.954259929",
        "-0.956393510",
        "0.921622976"
      ]
    ],
    [
      [
        "0.301834399",
        "0.099372180",
        "-0.049318023",
        "-0.558519295",
        "-0.386983422"
      ],
      [
        "0.289581495",
        "-0.612922472",
        "0.317864153",
        "-0.636910861",
        "0.796750295"
      ]
    ],
    [
      [
        "0.708802108",
        "-0.650517803",
        "-0.340338137",
        "-0.510360282",
        "0.740680006"
      ],
      [
        "0.821235458",
        "0.243142825",
        "0.159411406",
        "-0.221287881",
        "0.815068355"
      ]
    ],
    [
      [
        "0.441029052",
        "0.710663972",
        "0.444327086",
        "0.983272278",
        "-0.175913369"
      ],
      [
        "0.071916886",
        "-0.549886111",
        "0.477459460",
        "-0.705192672",
        "-0.662644119"
      ]
    ],
    [
      [
        "0.218913250",
        "0.278190624",
        "0.196509362",
        "0.540248617",
        "0.927502986"
      ],
      [
        "0.135005790",
        "-0.996049352",
        "0.252658415",
        "0.714983867",
        "-0.315878783"
      ]
    ],
    [
      [
        "0.540198432",
        "-0.625075898",
        "0.622546006",
        "0.244827033",
        "0.580979130"
      ],
      [
        "0.595317171",
        "-0.747330020",
        "-0.526345057",
        "0.464353334",
        "0.935659680"
      ]
    ],
    [
      [
        "0.252831208",
        "0.277731993",
        "-0.986309391",
        "-0.592886948",
        "-0.556645658"
      ],
      [
        "0.751939963",
        "0.359221910",
        "0.799494364",
        "0.511120064",
        "0.215709400"
      ]
    ],
    [
      [
        "0.711276366",
        "0.234666711",
        "-0.583435672",
        "0.186238641",
        "0.513600632"
      ],
      [
        "0.684734547",
        "0.154543195",
        "-0.521240071",
        "0.385823094",
        "-0.800782200"
      ]
    ],
    [
      [
        "0.471994910",
        "0.630808837",
        "0.343765731",
        "-0.206434092",
        "-0.630990775"
      ],
      [
        "0.677987027",
        "-0.532942363",
        "-0.157669037",
        "0.114639782",
        "-0.397892388"
      ]
    ],
    [
      [
        "0.365473180",
        "0.795085782",
        "-0.949173026",
        "0.756246733",
        "-0.031385624"
      ],
      [
        "0.305826281",
        "0.637140595",
        "0.800507424",
        "-0.939020824",
        "0.146515561"
      ]
    ],
    [
      [
        "0.627628639",
        "-0.754801459",
        "0.428646464",
        "-0.235819697",
        "-0.298037293"
      ],
      [
        "0.983666721",
        "0.167149199",
        "-0.651154452",
        "-0.630152442",
        "0.203735790"
      ]
    ]
  ]
}

提交格式与技术细节需要编写程序或准备 JSON 答案时再查看＋

子题参数

{
  "n": 12,
  "d": 5,
  "k": 2
}

当前第一名的答案

{
  "subspaces": [
    [
      [
        "0.638580726",
        "-0.442737934",
        "-0.744541137",
        "0.675667433",
        "-0.131600116"
      ],
      [
        "0.456108110",
        "-0.349357897",
        "-0.883927891",
        "-0.919138830",
        "-0.004274815"
      ]
    ],
    [
      [
        "0.414307950",
        "-0.162017071",
        "0.206788008",
        "0.369835077",
        "-0.299525310"
      ],
      [
        "0.840379191",
        "0.112683125",
        "0.954259929",
        "-0.956393510",
        "0.921622976"
      ]
    ],
    [
      [
        "0.301834399",
        "0.099372180",
        "-0.049318023",
        "-0.558519295",
        "-0.386983422"
      ],
      [
        "0.289581495",
        "-0.612922472",
        "0.317864153",
        "-0.636910861",
        "0.796750295"
      ]
    ],
    [
      [
        "0.708802108",
        "-0.650517803",
        "-0.340338137",
        "-0.510360282",
        "0.740680006"
      ],
      [
        "0.821235458",
        "0.243142825",
        "0.159411406",
        "-0.221287881",
        "0.815068355"
      ]
    ],
    [
      [
        "0.441029052",
        "0.710663972",
        "0.444327086",
        "0.983272278",
        "-0.175913369"
      ],
      [
        "0.071916886",
        "-0.549886111",
        "0.477459460",
        "-0.705192672",
        "-0.662644119"
      ]
    ],
    [
      [
        "0.218913250",
        "0.278190624",
        "0.196509362",
        "0.540248617",
        "0.927502986"
      ],
      [
        "0.135005790",
        "-0.996049352",
        "0.252658415",
        "0.714983867",
        "-0.315878783"
      ]
    ],
    [
      [
        "0.540198432",
        "-0.625075898",
        "0.622546006",
        "0.244827033",
        "0.580979130"
      ],
      [
        "0.595317171",
        "-0.747330020",
        "-0.526345057",
        "0.464353334",
        "0.935659680"
      ]
    ],
    [
      [
        "0.252831208",
        "0.277731993",
        "-0.986309391",
        "-0.592886948",
        "-0.556645658"
      ],
      [
        "0.751939963",
        "0.359221910",
        "0.799494364",
        "0.511120064",
        "0.215709400"
      ]
    ],
    [
      [
        "0.711276366",
        "0.234666711",
        "-0.583435672",
        "0.186238641",
        "0.513600632"
      ],
      [
        "0.684734547",
        "0.154543195",
        "-0.521240071",
        "0.385823094",
        "-0.800782200"
      ]
    ],
    [
      [
        "0.471994910",
        "0.630808837",
        "0.343765731",
        "-0.206434092",
        "-0.630990775"
      ],
      [
        "0.677987027",
        "-0.532942363",
        "-0.157669037",
        "0.114639782",
        "-0.397892388"
      ]
    ],
    [
      [
        "0.365473180",
        "0.795085782",
        "-0.949173026",
        "0.756246733",
        "-0.031385624"
      ],
      [
        "0.305826281",
        "0.637140595",
        "0.800507424",
        "-0.939020824",
        "0.146515561"
      ]
    ],
    [
      [
        "0.627628639",
        "-0.754801459",
        "0.428646464",
        "-0.235819697",
        "-0.298037293"
      ],
      [
        "0.983666721",
        "0.167149199",
        "-0.651154452",
        "-0.630152442",
        "0.203735790"
      ]
    ]
  ]
}

提交 subspaces：恰好 n 个子空间，每个是 2 个长度为 d 的基向量（十进制字符串，[-1, 1]）。基的选择不影响分数：同一个平面的任何基得同一个分。 · 验证器 v1.0.0

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